Exact analytical approach for free longitudinal vibration of nanorods based on nonlocal elasticity theory from wave standpoint

In this paper, free longitudinal vibration of nanorods is investigated from the wave viewpoint. The Eringen’s nonlocal elasticity theory is used for nanorods modelling. Wave propagation in a medium has a similar formulation as vibrations and thus,  it can be used to describe the vibration behavior. Boundaries reflect the propagating waves after incident. Firstly, the governing quation of nanorods longitudinal vibration based on the Eringen’s nonlocal elasticity theory is derived. Secondly, the propagation matrix for nanorod waveguide is derived and then the reflection atrix for spring boundary condition is calculated. The relations between amplitudes of propagation and reflection waves in the waveguide dominant are then combined in a matrix form format to set up a laconic efficient method for free axial vibration analysis of nanorods. The exact analytical solution for arbitrary boundary conditions natural frequencies is derived. To validate this approach, the exact solutions of special boundary conditions cases (clamped-clamped and clamped-free) are used. At the end, the effect of nonlocal parameter on the natural frequencies and boundary stiffness for arbitrary boundary condition is discussed